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Large sample distribution of the likelihood ratio test for normal mixtures

Author

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  • Chen, Hanfeng
  • Chen, Jiahua

Abstract

This article concerns with the problem of testing whether a mixture of two normal distributions with bounded means and specific variance is simply a pure normal. The large sample behavior of the likelihood ratio test for the problem is carefully investigated. In the case of one mean parameter, it is shown that the large sample null distribution of the likelihood ratio test statistic is the squared supremum of a Gaussian process with zero mean and explicitly given covariances. In the case of two mean parameters, both the simple and composite hypotheses of normality are considered. Under the simple null hypothesis, the large sample null distribution is found to be an independent sum of a chi-square variable and the squared supremum of another Gaussian process whose covariance structure is slightly different from the one mean parameter case, while under the composite null hypothesis, the chi-square term is absent.

Suggested Citation

  • Chen, Hanfeng & Chen, Jiahua, 2001. "Large sample distribution of the likelihood ratio test for normal mixtures," Statistics & Probability Letters, Elsevier, vol. 52(2), pages 125-133, April.
  • Handle: RePEc:eee:stapro:v:52:y:2001:i:2:p:125-133
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    Citations

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    Cited by:

    1. Song Qin, Yong & Smith, Bruce, 2006. "The likelihood ratio test for homogeneity in bivariate normal mixtures," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 474-491, February.
    2. Hung-Chia Chen & James J. Chen, 2016. "Hybrid Mixture Model for Subpopulation Identification," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 8(1), pages 28-42, June.
    3. Shaoting Li & Jiahua Chen & Jianhua Guo & Bing-Yi Jing & Shui-Ying Tsang & Hong Xue, 2015. "Likelihood Ratio Test for Multi-Sample Mixture Model and Its Application to Genetic Imprinting," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(510), pages 867-877, June.
    4. Jiaying Gu & Roger Koenker & Stanislav Volgushev, 2017. "Testing for homogeneity in mixture models," CeMMAP working papers CWP39/17, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    5. Maciejowska, Katarzyna, 2013. "Assessing the number of components in a normal mixture: an alternative approach," MPRA Paper 50303, University Library of Munich, Germany.
    6. Garel, Bernard, 2007. "Recent asymptotic results in testing for mixtures," Computational Statistics & Data Analysis, Elsevier, vol. 51(11), pages 5295-5304, July.

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